A324744 -id:A324744 - cp's OEIS frontend

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376

Offset: 1

Gus Wiseman, Mar 14 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(1) = 1 through a(6) = 20 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {2,6}
                       {2,4,5}  {3,4}
                                {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {4,5,6}
                                {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 1..100

The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.

Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(0) = 1 through a(10) = 6 strict integer partitions:
  ()  (1)  (2)  (3)  (4)    (5)  (6)    (7)    (8)    (9)    (10)
                     (3,1)       (4,2)  (4,3)  (6,2)  (5,4)  (6,4)
                                 (5,1)  (5,2)  (7,1)  (6,3)  (7,3)
                                        (6,1)         (7,2)  (8,2)
                                                             (9,1)
                                                             (6,3,1)

The subset version is A324738. The non-strict version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713.
Cf. A324744, A324750, A324755, A324760.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 8, 16, 35, 74, 166, 367, 831, 1878, 4299, 9857, 22775, 52777, 122957, 287337

Offset: 1

Gus Wiseman, Mar 18 2019

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
Also the number of finitary sets with n brackets where no element is also a subset. For example, the a(7) = 8 sets are (o = {}):
  {{{{{{o}}}}}}
  {{{{o,{o}}}}}
  {{{o,{{o}}}}}
  {{o,{{{o}}}}}
  {{o,{o,{o}}}}
  {{{o},{{o}}}}
  {{o},{{{o}}}}
  {{o},{o,{o}}}

			The a(1) = 1 through a(8) = 16 rooted identity trees:
  o  ((o))  (((o)))  ((o(o)))   (((o(o))))   ((o)(o(o)))    (((o))(o(o)))
                     ((((o))))  ((o((o))))   ((o(o(o))))    (((o)(o(o))))
                                (((((o)))))  ((((o(o)))))   (((o(o(o)))))
                                             (((o)((o))))   ((o)((o(o))))
                                             (((o((o)))))   ((o)(o((o))))
                                             ((o)(((o))))   ((o((o(o)))))
                                             ((o(((o)))))   ((o(o)((o))))
                                             ((((((o))))))  ((o(o((o)))))
                                                            (((((o(o))))))
                                                            ((((o)((o)))))
                                                            ((((o((o))))))
                                                            (((o)(((o)))))
                                                            (((o(((o))))))
                                                            ((o)((((o)))))
                                                            ((o((((o))))))
                                                            (((((((o)))))))

Cf. A000081, A290760, A304360, A306844, A317787.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324767, A324770, A324771, A324838, A324840, A324844, A324846.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]

A324845 Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of terms together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))

Gus Wiseman, The sequence of rooted trees whose Matula-Goebel numbers belong to A324845.

Cf. A000081, A007097, A290822, A306844, A318186.

Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324842, A324844, A324846, A324847, A324849.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]

cp's OEIS Frontend

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

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A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

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A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

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A324845 Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

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Mathematica